Abstract
In the present work, we report a detailed description of the symmetry properties of the eight-atomic molecule ethane, with the aim of facilitating the variational calculations of rotation-vibration spectra of ethane and related molecules. Ethane consists of two methyl groups CH 3 where the internal rotation (torsion) of one CH 3 group relative to the other is of large amplitude and involves tunnelling between multiple minima of the potential energy function. The molecular symmetry group of ethane is the 36-element group G 36 , but the construction of symmetrised basis functions is most conveniently done in terms of the 72-element extended molecular symmetry group G 36 (EM). This group can subsequently be used in the construction of block-diagonal matrix representations of the ro-vibrational Hamiltonian for ethane. The derived transformation matrices associated with G 36 (EM) have been implemented in the variational nuclear motion program TROVE (Theoretical ROVibrational Energies). TROVE variational calculations are used as a practical example of a G 36 (EM) symmetry adaptation for large systems with a non-rigid, torsional degree of freedom. We present the derivation of irreducible transformation matrices for all 36 (72) operations of G 36 (M) (G 36 (EM)) and also describe algorithms for a numerical construction of these matrices based on a set of four (five) generators. The methodology presented is illustrated on the construction of the symmetry-adapted representations both of the potential energy function of ethane and of the rotation, torsion and vibration basis set functions.
Highlights
In variational calculations of molecular rotation-vibration energies and wavefunctions, a matrix representation of the molecular rotation-vibration Hamiltonian, constructed in terms of suitable basis functions, is diagonalised numerically
We report an analogous analysis for the molecular symmetry (MS) group G36 and the extended molecular symmetry (EMS) group G36 (EM) [1]
To deal with the double-valuedness of (τ, χ), we extend the symmetry description in the manner first introduced by Hougen [43], by extending G36 to the extended molecular symmetry group G36 (EM)
Summary
In variational calculations of molecular rotation-vibration energies and wavefunctions, a matrix representation of the molecular rotation-vibration Hamiltonian, constructed in terms of suitable basis functions, is diagonalised numerically. It is well known that this type of calculation can be facilitated by the introduction of symmetrised basis functions, i.e., basis functions that generate irreducible representations of a symmetry group for the molecule in question (see, for example, [1]). In a recent paper [2], we described an example of such a symmetry adaption, in that we presented character tables and irreducible representation transformation matrices for Dnh groups with arbitrary n-values. With these results, we could practically utilise the linear-molecule symmetry properties described by the D∞h point group in numerical calculations with the variational nuclear motion program TROVE [3], the acetylene molecule 12 C2 H2 serving as an example. The matrix groups obtained are used for symmetrising the potential energy surface and the rotation-vibration basis functions for ethane
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